It is very convenient to use the shorter notation
Note that whenever the symbol >=< is used there are three parallel statements being made.
The four magnitudes do not all have to be of the same kind, but the first pair w and x need to be of the one kind, and the second pair y and z of one kind, either the same kind as that of w and x or a different kind. Perhaps the best illustration of these definitions comes from proposition VI.1 in which Euclid first uses them to construct a proportion.
According to Definition 5, in order to show the ratios are the same, Euclid takes any one multiple of BC and ABC (which he illustrates by taking three times each), and any one multiple of CD and ACD (which he also illustrates by taking three times each). Then he proceeds to show that the former equimultiples, namely HC and CL, alike exceed, are alike equal to, or alike fall short of, the latter equimultiples, namely, AHC and ACL.
Symbolically, in order to prove BC : CD = ABC : ACD, Euclid proves for any numbers n and m that the line n BC is greater, equal, or less than the line m CD when the triangle n ABC is greater, equal, or less than the triangle m ACD. We will abbreviate this condition symbolically as
Note that in order to check this condition, it is only necessary to compare lines to lines and planar figures to planar figures. To see how Euclid does this, refer to VI.1.
Using modern concepts and notations, we can more easily see what the general definition of equality of two magnitudes means. If we treat ratios as real numbers, the a proportion such as the one described above, BC : CD = ABC : ACD, means that the ratio BC : CD compares to all numerical ratios (that is, rational numbers) m/n the same way that ABC : ACD does. Another way of saying this is that equality of two real numbers is determined by their relation to all rational numbers. This is often expressed by saying that the set of rational numbers is dense in the set of real numbers.
Of course, Euclid did not have what modern mathematicians call real numbers. Indeed, there is an ontological difference between real numbers and Euclid’s ratios. Some real numbers are not ratios of the magnitudes of any kind mentioned in the Elements.
First, reflexivity. Is it the case for any pair of magnitudes of the same type A and B that A and B are in the same ratio as A and B? That means for any numbers m and n,
That is trivial.
Second, symmetry. Is it the case that if A and B are in the same ratio as C and D, then C and D are in the same ratio as A and B? The first says
while the second says
This can be shown using the law of trichotomy for magnitudes. (Suppose nC > mD. If nA is not greater than mB, then it is less or equal, but then nC is less or equal to mD, contradicting nC > mD. etc.) Euclid missed symmetry, but he uses it very frequently.
Third, transitivity. Euclid states this explicitly in proposition V.11. The proof relies only on the definition.
Thus, proportion is an equivalence relation.
A more fundamental question is “do ratios exist?” Are they some kind of mathematical object like numbers and magnitudes? The Elements do not require it. Instead, proportion is a relation held between one pair of magnitudes and another pair of magnitudes. Yet it is very easy to read Book V as though ratios are mathematical objects of some abstract variety. And it’s easy to read “A and B have the same ratio as C and D” as saying that the ratio A : B is the same ratio as C : D.
Not every relation allows that reading, but equivalence relations do, and proportion is an equivalence relation.
The philosophical questions “do ratios exist?” and “is a proportion equality of ratios?” can be converted to the question “why do equivalence relations create entities?” or a little more conservatively, “why do equivalence relations allow us to think and act as if the entities exist?”
It is hard to imagine that Euclid did not think of ratios as things and proportions as equalities, especially since the next definition defines when one ratio is larger than another. Perhaps he did, but continued to write noncommittally.
Proportions are written as equalities in the Guide.
which is just the definition of proportion when m and n are both equal to 1. See the Guide for Proposition V.14 for further comments.